Let $G$ be a group scheme (for instance, over $k$ a field of characteristic 0).
Let $e$ be its unit.
I denote by $O_G$ the structural sheaf of $G$.

Let $D_e : O_{G,e} \to k$ a derivation.

I would like to get directly (ie, without any consideration about the cotangent bundle, or some canonical isomorphisms...) a derivation $D : O_G\to O_G$ that extends $D_e$, and which is compatible with the action of $G$. That is, I would like to get this derivation by the mean of the multiplication map : $m : G \times G \to G$, etc., etc.

I have guessed this question would not be difficult, and would only be a matter of technics, but I can't manage to do it.

You interpret your derivation $D_e$ as a distribution on $G$ supported in $e$, and then your derivation $D$ is the convolution with $D_e$ with respect to $m$. I.e., take your local function on $G$, compose it with $m$ to obtain a local function on $G\times G$, and apply $D_e$ along one of the arguments.